Finite Group Databasebeta

← #16828328

Subgroups — #16828328

Order 3712.

470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745277C211yesmaximal, characteristic
2#14271642C211yesmaximal, characteristic
3#14271639C211yesmaximal, characteristic, complemented
4#14271269C211yesmaximal, characteristic, complemented
5#14271268C211yesmaximal, characteristic, complemented
6#14271264C211yesmaximal, characteristic, complemented
7#14271263C211yesmaximal, characteristic, complemented
8C4×Dic116C211yesmaximal, characteristic, complemented
9#14271255C211yesmaximal, characteristic, complemented
10#14271254C211yesmaximal, characteristic, complemented
11#14271202C211yesmaximal, characteristic, complemented
12#14271156C211yesmaximal, characteristic
13#103×D29C211yesmaximal, characteristic, complemented
14#14271117C211yesmaximal, characteristic, complemented
15C4×SD16×C29C211yesmaximal, characteristic, complemented
16C2×Q8×D29C2211yescharacteristic
17C2×#90807C2211yescharacteristic
18C2×SD16×C29C2211yescharacteristic
19C4×Q8×C29C2211yescharacteristic
20C4×D4×C29C2211yescharacteristic
21Q8×Dic29C2211yescharacteristic
22C2×#90785C2211yescharacteristic
23D4×Dic29C2211yescharacteristic
24C2×#90783C2211yescharacteristic
25C4×C22⋊D29—21no—
26C2×#90779—21no—
27#11745277C418yescomplemented
28C2×Dic116C2211yescharacteristic
29C2×#90772C2211yescharacteristic
30C2×C8×D29C2211yescharacteristic
31#11745254—21no—
32#11745254C2211yescharacteristic
33C4⋊C4×D29C2211yescharacteristic
34#11745249—21no—
35#11745249C2211yescharacteristic
36#11745249C2211yescharacteristic
37#11745243—21no—
38#11745239—21no—
39#11745234—21no—
40#11745234C2211yescharacteristic
41C4×Dic58—21no—
42C4×Dic58C2211yescharacteristic
43C4×Dic58C2211yescharacteristic
44C29×#106C2211yescharacteristic
45C29×#103C2211yescharacteristic
46C29×#102C2211yescharacteristic
47C4×C8×C29C2211yescharacteristic
48#11745183C2211yescharacteristic
49#11745180C2211yescharacteristic
50#11745166C2211yescharacteristic
51#11745164C2211yescharacteristic
52#11745164C2211yescharacteristic
53#11745163C2211yescharacteristic
54C8×Dic29C2211yescharacteristic
55#11745158C2211yescharacteristic
56#11745158C2211yescharacteristic
57#11745156C2211yescharacteristic
58#11745155C2211yescharacteristic
59#11745151C2211yescharacteristic
60C2×Q8×C29C2311yescharacteristic
61C2×D4×C29C2311yescharacteristic
62C22×C4×C29—21no—
63C2×C22⋊D29—21no—
64C22×Dic29—21no—
65C22×Dic29—21no—
66Q8×D29—22no—
67Q8×D29C2×C414yescomplemented
68#90807—22no—
69#90807C2×C414yes—
70C2×C4×D29—21no—
71C2×C4×D29—21no—
72C2×C4×D29C2311yescharacteristic
73C2×Dic58—21no—
74C2×Dic58C2311yescharacteristic
75C2×Dic58C2311yescharacteristic
76SD16×C29C2×C414yescomplemented
77C2×C8×C29C2311yescharacteristic
78C2×C8×C29C2311yescharacteristic
79C4⋊C4×C29—21no—
80C4⋊C4×C29C2311yescharacteristic
81C4⋊C4×C29C2311yescharacteristic
82C22⋊C4×C29—21no—
83C42×C29—21no—
84C42×C29C2311yescharacteristic
85C22⋊Dic29—21no—
86#90785C2×C414yescomplemented
87#90783C2×C414yescomplemented
88#90781—21no—
89#90781D411yescharacteristic
90#90781D411yescharacteristic
91C4⋊Dic29—21no—
92C4⋊Dic29C2311yescharacteristic
93C4⋊Dic29C2311yescharacteristic
94#90779—21no—
95#90779—21no—
96#90779—21no—
97#90779—21no—
98#90779—21no—
99#90779D411yescharacteristic
100#90779D411yescharacteristic
101C4×Dic29—21no—
102C4×Dic29—21no—
103C4×Dic29—21no—
104C4×Dic29—21no—
105C4×Dic29—21no—
106C4×Dic29—21no—
107C4×Dic29C2311yescharacteristic
108C2×#6597C2311yescharacteristic
109C2×#6597C2311yescharacteristic
110Dic116C2×C414yescomplemented
111#90772C2×C414yescomplemented
112C8×D29—22no—
113C23×C29—21no—
114C22×D29C2×D411yescharacteristic
115Q8×C29—21no—
116Q8×C29C22×C412yes—
117D4×C29—21no—
118D4×C29C22×C412yes—
119C2×C4×C29—21no—
120C2×C4×C29—21no—
121C2×C4×C29—21no—
122C2×C4×C29—81no—
123C2×C4×C29C2×D411yescharacteristic
124C2×C4×C29C2×D411yescharacteristic
125C2×C4×C29C2411yescharacteristic
126C22⋊D29—24no—
127C2×Dic29—21no—
128C2×Dic29—21no—
129C2×Dic29—24no—
130C2×Dic29—21no—
131C2×Dic29—21no—
132C2×Dic29—81no—
133C2×Dic29C2×D411yescharacteristic
134C2×Dic29C2×D411yescharacteristic
135C2×Dic29C2×D411yescharacteristic
136C4×D29—24no—
137C4×D29—81no—
138C4×D29C22×C412yes—
139C4×D29#4112yes—
140Dic58—21no—
141Dic58—21no—
142Dic58—24no—
143Dic58C22×C412yes—
144Dic58C22×C412yes—
145C8×C29—21no—
146C8×C29C22×C412yes—
147#6597—21no—
148#6597C22×C412yes—
149#2947—291nomaximal, Sylow
150C22×C29—22no—
151C22×C29—41no—
152C22×C29—41no—
153C22×C29C22×D411yescharacteristic
154C2×D29—21no—
155C2×D29—21no—
156C2×D29C4×D412yes—
157C4×C29—22no—
158C4×C29—21no—
159C4×C29—21no—
160C4×C29—41no—
161C4×C29—41no—
162C4×C29C23×C411yescharacteristic
163C4×C29C2×#4111yescharacteristic
164Dic29—22no—
165Dic29—22no—
166Dic29—21no—
167Dic29—21no—
168Dic29—41no—
169Dic29—41no—
170Dic29C4×D412yes—
171C2×#137—291no—
172#520—291no—
173#519—291no—
174#441—291no—
175#441—291no—
176#440—291no—
177#440—291no—
178C4×Q16—291no—
179C4×Q16—291no—
180C4×SD16—291no—
181C4×SD16D2911yescharacteristic, complemented
182#428—291no—
183#416—291no—
184C2×#103—291no—
185#405—291no—
186C2×C29—42no—
187C2×C29C2×C4×D411yescharacteristic
188C2×C29#57711yescharacteristic
189C2×C29#57811yescharacteristic
190D29—22no—
191C2×#41—291no—
192C22×Q8—291no—
193#137—298no—
194C2×Q16—291no—
195C2×Q16—291no—
196C2×SD16—291no—
197C2×SD16C2×D2911yescharacteristic
198C2×M16—291no—
199C22×C8—291no—
200C4×Q8—291no—
201C4×Q8—291no—
202C4×Q8—291no—
203C4×Q8—291no—
204C4×Q8—291no—
205C4×Q8—581no—
206C4×Q8—581no—
207C4×Q8C2×D2911yescharacteristic
208C4×D4—291no—
209C4×D4—581no—
210C4×D4—581no—
211C4×D4C2×D2911yescharacteristic
212#117—291no—
213#117—291no—
214#117—581no—
215#117—581no—
216#117—581no—
217C2×C4⋊C4—291no—
218C2×C4⋊C4—581no—
219#107—291no—
220#107—291no—
221#106—291no—
222#106C2×D2911yescharacteristic
223#103—291no—
224#103—291no—
225#103—291no—
226#103—291no—
227#103—291no—
228#103C2×D2911yescharacteristic
229#102—291no—
230#102C2×D2911yescharacteristic
231#97—291no—
232#97—291no—
233C4×C8—291no—
234C4×C8C2×D2911yescharacteristic
235C29#294711yescharacteristic, complemented, Sylow
236#41—294no—
237#41—582no—
238C2×Q8—294no—
239C2×Q8—291no—
240C2×Q8—291no—
241C2×Q8—581no—
242C2×Q8—582no—
243C2×Q8C22×D2911yescharacteristic
244C2×D4—581no—
245C2×D4C22×D2911yescharacteristic
246C22×C4—21no—
247C22×C4—291no—
248C22×C4—581no—
249C22×C4—581no—
250C22×C4—581no—
251C22×C4—581no—
252Q16—294no—
253Q16—294no—
254SD16—294no—
255SD16C4×D2914yescomplemented
256M16—294no—
257C2×C8—291no—
258C2×C8—291no—
259C2×C8—582no—
260C2×C8C22×D2911yescharacteristic
261C2×C8C22×D2911yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—291no—
264C4⋊C4—291no—
265C4⋊C4—291no—
266C4⋊C4—291no—
267C4⋊C4—581no—
268C4⋊C4—581no—
269C4⋊C4—581no—
270C4⋊C4—581no—
271C4⋊C4—581no—
272C4⋊C4—581no—
273C4⋊C4C22×D2911yescharacteristic
274C4⋊C4C22×D2911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—291no—
277C22⋊C4—291no—
278C22⋊C4—581no—
279C22⋊C4—581no—
280C42—21no—
281C42—291no—
282C42—581no—
283C42—581no—
284C42—581no—
285C42—581no—
286C42—581no—
287C42—581no—
288C42C22×D2911yescharacteristic
289C23—21no—
290C23—291no—
291Q8—21no—
292Q8—292no—
293Q8—292no—
294Q8—584no—
295Q8—581no—
296Q8—581no—
297Q8C2×C4×D2912yes—
298D4—21no—
299D4—584no—
300D4C2×C4×D2912yes—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—291no—
306C2×C4—292no—
307C2×C4—292no—
308C2×C4—291no—
309C2×C4—291no—
310C2×C4—584no—
311C2×C4—584no—
312C2×C4—581no—
313C2×C4—581no—
314C2×C4—581no—
315C2×C4—581no—
316C2×C4—2321no—
317C2×C4—2321no—
318C2×C4D4×D2911yescharacteristic
319C2×C4D4×D2911yescharacteristic
320C2×C4C23×D2911yescharacteristic
321C8—21no—
322C8—292no—
323C8—581no—
324C8C2×C4×D2912yes—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—292no—
329C22—581no—
330C22—581no—
331C22C2×D4×D2911yescharacteristic
332C4—22no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—292no—
338C4—582no—
339C4—582no—
340C4—581no—
341C4—581no—
342C4—1161no—
343C4—1161no—
344C4C22×C4×D2911yescharacteristic
345C4#41×D2911yescharacteristic
346C2—42no—
347C2—582no—
348C2C4×D4×D2911yescharacteristic
349C2#1427196911yescharacteristic
350C2#1427197711yescharacteristic
subgroupG/H
centerC22C2×D4×D29
commutatorC4×C29C23×C4
FrattiniC2×C4C23×D29
FittingC4×SD16×C29C2
socleC22×C29C22×D4
prime radicalC2×#90807C22
cosocleC2×C4×C29C24
hypercenterC4×SD16D29
hypocenterC29#2947
LCSG ▷ C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #14271639C2semidirect
4. #14271269C2semidirect
5. #14271268C2semidirect
6. #14271264C2semidirect
7. #14271263C2semidirect
8. C4×Dic116C2semidirect
9. #14271255C2semidirect
10. #14271254C2semidirect
11. #14271202C2semidirect
13. #103×D29C2semidirect
14. #14271117C2semidirect
15. C4×SD16×C29C2semidirect
27. #11745277C4semidirect
67. Q8×D29C2×C4semidirect
76. SD16×C29C2×C4semidirect
86. #90785C2×C4semidirect
87. #90783C2×C4semidirect
110. Dic116C2×C4semidirect
111. #90772C2×C4semidirect
181. C4×SD16D29semidirect
235. C29#2947semidirect
255. SD16C4×D29semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117452772
#142716422
#142716392
#142712692
#142712682
#142712642
#142712632
C4×Dic1162
#142712552
#142712542
#142712022
#142711562
#103×D292
#142711172
C4×SD16×C292
#294729